A graph $(\kappa, E)$ is highly homogeneous if all its restrictions on complements of sets of cardinality less than $\kappa$ are homogeneous. We investigate for which cardinals $\theta < \lambda \leq \kappa$ does hold that for every coloring $c:[\kappa]^2\to \theta$ there exists $A$ subset of $\kappa$ of cardinality $\lambda$ and a color $i\in\theta$ such that the graph $(A, c^{-1}(i)\cap [A]^2)$ is highly homogeneous.

Joint work with J. Bergfalk and S. Shelah.

Joint work with J. Bergfalk and S. Shelah.